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From: sthomas@decan.gate.net (S. F. Thomas)
Subject: Re: Fuzzy theory or probability theory?
Message-ID: <1994Nov30.212130.5001@decan.gate.net>
Organization: Decision Analytics, Inc.
Date: Wed, 30 Nov 1994 21:21:30 GMT
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Radford Neal (radford@cs.toronto.edu) wrote:
: 
: To take an actual real-world situation, consider the problem of a 
: police officer who has interviewed a witness to a crime.  The witness
: described the perpetrator as "tall".  How should we formalize how the
: officer should handle this information?
: 
: I think it is clear that what the police officer needs here is a 
: likelihood function - for any actual height (plus any other characteristics
: that might influence subjective assessment of height), the officer
: needs to know the PROBABILITY that the witness would describe the
: such a perpetrator as "tall".  This way if two suspects are later found,
: one can compare how well they each fit the testimony of the witness.
: 

I had the same insight some 15 years ago, and it was the key to my
Ph.D dissertation "A theory of semantics and possible inference, with
application to decision analysis", University of Toronto, 1979.
I left academia right after that, but worked off and on (mostly off)
to refine the dissertation, coming up with a manuscript entitled 
"Fuzziness and Probability", February, 1992.  I sent it at one point
to John Wiley, who never responded, then I got sidetracked by 
various life changes...  This thread has encouraged me to dust off 
that manuscript and try again in earnest to get it published...

Here are some excerpts from that manuscript bearing on the subject of
this thread:

From Chap. III, "Fuzzy Set Theory of Semantics":

"I have indicated in Chap. I that fuzzy set theory may
contribute to a reformulated likelihood theory of inference.
Unfortunately, the Zadehian theory of fuzzy sets does not, in its
fundamentals, allow the intimate relation between fuzziness and
probability required for such an endeavor.  I therefore want in the
present chapter first to review briefly the Zadehian development of
fuzzy set theory, indicating the points of disagreement, and then to
develop the alternative which I propose.

 ...

Zadeh (1975) has taken the position that the notion of grade
of membership is merely a subjective estimation of the extent to which
any given element may be said to belong to any fuzzy set in question.
On this view it is difficult to decide whether the set of tall men,
for example, could not simply be represented as a Bayesian subjective
probability distribution over the space of height values.  It is also
difficult to establish any particular set of combination rules.  The
minimum-maximum rules of the Zadehian calculus have intuitive appeal,
but lead to debatable consequences.  In particular, the
self-contradiction law, the law of contradiction, and the law of excluded
middle are violated.  Should they be, and if not, what other rules of
combination may we substitute that would restore these laws.  Zadeh
has also taken the position that the concepts of probability and
fuzziness are distinct, raising the question what kind of statistical
methods could one logically apply to establish either a membership
function, or any particular rules of combination which one might care
to propose.

As I have indicated previously, I find it difficult to accept
the idea that membership functions may be entirely subjective.  If I
were to assert that 6ft. is short for a man, I think one would be
entitled to question whether I were a competent speaker of the English
language.  Thus there is an element of convention in the meaning of
words in a language.  As I have tried to point out in the previous
chapter, if a convention exists, then one's subjective estimation
cannot be the whole story -- there must be an external reality out
there regarding language use susceptible of objective
characterization.

Like Watanabe (1978), I also find it difficult to accept the
result of the Zadehian min-max calculus that the fuzzy term "tall and
not tall" should be anything less than the logical absurdity, as the law of
contradiction requires.  One would lose all credibility as a witness
in court if one were to testify that the burglar was "tall, but not
tall".  The fuzziness of the term tall is not sufficient, in my view,
to persuade a jury that such a contradiction could have positive
meaning within the English language convention.  Similarly, the law of
excluded middle requires that the disjunction "tall or not tall"
should be the constant tautology.  Again, I am not persuaded that the
fuzziness of the term tall is sufficient to justify the result of the
min-max calculus under which not all elements of the universe have
full membership, tautologically, in this disjunction.  Finally, the
law of self-contradiction requires that any term which implies its own
negation must be the logical absurdity.  Under the min-max calculus,
any term, not necessarily the absurdity, whose membership function is
everywhere less than half must imply its own negation.

In what follows I depart from the Zadehian fuzzy set theory
first at the philosophical level.  I take the membership function to
represent a usage convention which may in principle be objectively
determined, using statistical methods.  Proceeding from this basic
assumption, the concepts of probability and fuzziness may indeed be
distinguished, but the concept of fuzziness is derived from that of
probability, in almost exactly the same fashion that the Fisherian
concept of likelihood derives from probability.  And in the same way
that the concepts of probability and likelihood are distinct, the
concepts of probability and fuzziness are distinct, though related
concepts.  Coincidentally, it turns out that proceeding from such an
assumption makes the min-max calculus quite simply untenable as a set
of universal rules.  It becomes clear that other rules are sometimes
appropriate, and it forces one to address the issue when does which
apply.  In so doing, the law of contradiction and the law of excluded
middle are upheld as a matter of necessity, a happy result if one is
disposed to accept these as having positive empirical significance as
semantic law.  The law of self-contradiction does not hold, as in the
Zadehian calculus, if one takes the containment relation between fuzzy
sets as representing the implication relation.  If, however, as in
fact is necessary in the present development, the rule of implication
is defined with reference to possibility distributions rather than
membership functions, and possibility distributions, like likelihood
functions, are unique only up to similarity transformations, then the
law of self-contradiction may be restored.

This result, and the restoration of the laws of contradiction
and of excluded middle, are happy byproducts, however, rather than
starting objectives.  My basic goal is rather to harmonize the
essential truth of the fuzzy set concept with the essential truth of
the concept of probability, and to try to sort out the respective
limits of application of the two concepts in the representation of
uncertainty."

: I fail to see how fuzzy logic provides any useful guide here, except 
: to the extent that it might be replicating this likelihood function,
: which is based on conventional probability.
: 

Another excerpt, relating now to fuzzy logic, from
Chap. V, "Possibilistic Deductive Inference":

"Deductive inference is concerned with arguing 
from premises to conclusions. In the present 
context, deductive inference is distinct from 
inductive inference in that while the latter
is concerned with characterizing what a 
speaker could mean by what he says, deductive 
inference is concerned with reformulating and/or 
rearranging and/or simplifying what a speaker may have 
said, as a preliminary perhaps to asking what the 
speaker could mean. For example, suppose a speaker 
S asserts "If John is rich then John is happy", and 
in addition asserts "John is rich", then without 
asking what S may mean either by "rich" or "happy" 
we may deduce that what S asserts is equivalent to 
saying "John is rich and happy", prior to the 
inductive problem of characterizing John's wealth 
and happiness on the basis of S's assertion. 
Deductive inference is therefore usually concerned 
more with form than with meaning. This is not to 
deny the major achievement of logic which is to 
lend clarity to a confused muddle of premises by 
drawing out, through purely formal re-structurings, 
a simple declarative statement regarding an unknown 
of interest. As Suppes (1957, p.68) has said,
"only a meager theory of meaning is needed to apply 
the axiomatic method". Our concern in the present 
development is to take a fairly elaborate theory of 
meaning from which a system of deductive inference 
relying not on form but on semantic content could 
be developed.

In this program, the viewpoint is a little 
different from that of Zadeh (1977), who saw
approximate reasoning as concerned with the 
"deduction of possibly imprecise conclusions from a 
set of imprecise premises". Traditional logic, by 
restricting itself to form rather than meaning 
already allows us to reach imprecise conclusions 
from imprecise premises, a fact that is exemplified 
by the example previously considered:

   If one is rich, then one is happy - Premise (Theory 1)

   John is rich                      - Premise (observation)

   Therefore, John is happy          - Conclusion.


Here both premises are imprecise, as is the 
conclusion, even though the rule of deduction which 
has been applied is quite exact, relying only on 
the logical form  [(P -> Q) /\ P] -> Q  in the standard
logic notation. What we now propose to do is to 
take meaning as primary, and to allow deductive 
inferences to be validly drawn whenever meaning is 
preserved, in a sense to be made clear very 
shortly. From this viewpoint, the appellation 
"approximate" in "approximate reasoning" would 
refer not so much to the rules of logic or of 
reasoning involved, but to the nature of the 
assertions involved, which may in general be fuzzy.
Traditional logic in dealing primarily with form, 
proceeds almost entirely on the semantics of form, 
hardly at all on the semantics of content. Here we 
start with content and rules based on form emerge 
as a special case. "

:     Radford Neal

Hope this is of some use.  If there is interest, I may
just upload it to an ftp site.  Otherwise, if there are any
publishers out there who are interested, it can be made
available as a postscript, dvi, or laserjet file, or in hard-copy,
or as a g(t)roff manuscript using eqn, tbl, etc.

Cheers
S.F.Thomas
